Learning to say "no" is one of the most difficult skills for a poly person to acquire.
I was having a think about it last night, after chatting with a friend who is fairly new to poly and is finding herself being overwhelmed with offers. In my opinion, saying no is actually a set of three skills:
1) Knowing when you need to say "no"
2) Being able to actually say it, and
3) Saying it in a positive, or at least non-hurtful, way.
Each one is a distinct skill, and it's possible to have one or more of them and lack others. All of them are important, though, and generally your poly experience will be easier and happier if you can master them all. That said, while I have #1 down pretty well now, I'm still working on consistently applying #2, and #3 is still very hard.
Number one is all about self-awareness, and it's more than just recognising compatibility (or a lack thereof) with another person. It's also about knowing your own limitations, recognising how much time and energy a new romantic commitment will take up and deciding whether you can spare them. One of the hardest things is meeting someone with whom you have great chemistry, but realising that your life is just too full for another relationship, so you have to tell them "no".
Number two is about confidence and trust in yourself. Whether you have no interest in someone who is interested in you, or you feel the same as they do but can't initiate anything for other reasons, actually coming out with it and saying "no" can be very tough. I think the strongest incentive to urge you to speak your mind is thinking about the potential consequences if you keep silent. Imagine someone developing deep feelings for you that you can't reciprocate because you couldn't bit the bullet and tell them it couldn't go anywhere. Like many emotional issues, it's one of things that gets more painful the longer you leave it.
Number three is a tough one. Nobody who is interested in you wants to hear that you don't feel the same about them, and it's the kind of news which is going to sting a bit, no matter how well you phrase it. Unless you're a nasty person, causing other people pain is something you want to avoid. This may contribute to a failure to exercise skill #2 above: turning someone down hurts, and you don't want to hurt anyone, so you have trouble turning people down even when you know you should.
I think there is an art to letting people down gently, rather than a precise science, so no advice is universally applicable. To be honest, I am not a good one to give advice on developing skill #3 anyway: I don't often find myself pursued, so rarely need to let people down, gently or otherwise. The last time I had to do it, I don't think I did a very good job. I recognise that this is a skill I really need to work on.
My problem is that I have an aversion to conflict (hooray for childhood traumas) and tend avoid saying "no" for as long as possible. When I finally say it, it's often because I've been backed into a corner and forced to finally be fully honest. As a result, I'm feeling cornered and defensive, and not in the loving headspace needed to tell someone they can't have what they want but avoid hurting them more than can be avoided.
Sorry this blog post is a little unsatisfying - this is an issue I am still working on myself.
I thought I'd add my two cents to this, and in particular number three on the list above. I think I am somewhat qualified to add to this given I seem to have a small queue of men who would love to take me to bed with them (at least), and whom I've had to turn down.
I think the art of saying no, is when its said with respect for the other's feelings, which James has already stated above. And also with a genuine sense of appreciation for how hard it is to actually ask someone for more than a hug, and an appreciation of how flattering it is to be desired.
I had to learn this, by the way. I had a close friend who was an artist at saying no to people in ways that left them happy, if a little disappointed. I never got to overhear any of her conversations with people she said no to, but she'd tell me later and I never got an explanation from her as to what she did.
So then came along the day when I actually realised that I needed to say no to someone. A lovely angel of a man who many people find quite sexy, and with whom I didn't feel any chemistry. I emailed him and apologised for my lack of chemistry, given that I felt that I was letting the side down given how desired he was by everyone else.
I wasn't happy with the whole emailing thing, but given he lived in another city, it was the best option at the time.
Then, more offers from lovely people who I adore, but feel no chemistry with (I'm quite picky). Each time I told them how flattered I was, how much I appreciated them asking me, but given I didn't feel any chemistry with them, it'd be wrong/unfair on both of us to take it further than we have. I am still close with these people, I appear to have let them know that I do value them as friends and confidants and am not dismissive of their desires. The fact that I continue lightly flirting with them and being affectionate with them probably helps - but that was the way I was with them before they propositioned me.
One sweet gentleman (and he'd laugh if he read that description of himself), asked me on IM once when we were chatting if I was interested in more with him. I said that I loved chatting with him, but that was as far as I was willing to take it. He said that he thought that was the case and just wanted to check and that he loved how he could be incredibly suggestive with me and I'd just laugh and continue talking with him, because there had been some people who took offense at that.
So, respect the individual and their courage in asking or propositioning you. Thank them for asking you and say no gently. This clearly applies to people you like and know and want to remain friends with. There will always be people who don't listen to a subtle no, and for whom, "fuck no" or even "fuck off" is the only answer they might hear. Don't feel bad in using that in that instance. If you've tried to say no in all the other ways and not been heard, telling someone directly, even if it does hurt their feelings, isn't a bad thing. Its better to be safe than sorry after all.