I love brights and use them a lot...
I try to think about two things: proportion and shading. Proportion meaning that I think it's much harder to make a successful quilt when everything is bright. Maybe only 30% to 40% should be bright and the rest should be toned down a bit. That's still a bright quilt!
This is similar to making a successful pastel quilt - not everything should be soft and light - most often a pastel quilt needs some darker fabrics too (not all the way to black, just darker), even if they make up only 10% of the quilt - to really be a stand-out.
And then shading - I don't think it's easy to make a mostly toned-down quilt and then BAM add brights. it's easier if you add several fabrics that progressively get more bright until you get to your brights.
About backgrounds, I've used blacks, grays, tans, browns, and off-whites. I think blacks make your brights 'pop' more and tans and off-whites tone them down a bit. Grays are really fun. Blue-y cool grays make your hot brights pop more, and warmer grays tend to make your cooler brights pop. Your choice - you're in control!
I guess, Lady, if you wanted short, definitive answers, maybe I'm not the one... Opps, that didn't quite sound right to call you 'Lady'. =) I'm so sorry Anna! Anyway, I hope this helps, Anna!