It is difficult to deal with children's gifts that are obnoxious, but I can't defend getting rid of the toy if the recipient likes it unless it is supremely problematic, like limabean stated. My dd is only two, so after a while, she stops using certain toys. At that point, I might quietly slip it out of the house. I already cycle her toys because we have limited space in our apartment, so it is easy to filter through things. But as she gets older, this won't be an option in my mind. I was thinking then I might explain that we only have enough room for "x" amount of stuff. I also want to encourage a philanthropic spirit in her so that she willingly donates some belongings to others.
The best thing is to work to curb the gift giving before it happens. I usually tell people I would prefer pictures, or a letter, their favorite recipes or poems, books, or anything handmade.
So the kinds of gifts that get the treatment: large things, loud things, toxic things, books or videos that promote consumer lifestyle. For example, my MIL sent her a ton of battery operated animals in her first year. Then we asked her to stop. Anyhow, first I cycled them into the closets, then I donated them or traded them in at the kid consignment store.
Since then, Dh and I have been pretty straight forward about what we like (wooden puzzles, picture books, safe art supplies) We don't require new stuff, but we almost always receive new. And usually we pick a theme of sorts and let our families do there thing. This Christmas it was puzzles and books. For her birthday, art supplies and wooden food. Of course, we got plenty more that didn't fall into these categories. Now I feel that we are saturated with stuff so I might switch to another tactic for 2014: passes to things, like the children's museum, zoo, botanical gardens, aquarium; also dress up supplies, which can just be things that people have around their house like costume parts, scarves, hats, old glasses frames, masks etc. I was also thinking about camping gear or outside adventure stuff (sleeping bag, nature observation stuff, etc). As she gets older, I may ask for family to help build her savings account if they really really want to spend money. Or help save for a trip to come see them. But family members like giving actual objects!
As for stuff that I don't like? I re-gifted one thing to a white-elephant party. I re-gifted a pair of earrings to an aunt that I knew would like them. I donate, I give away to friends, or I use the gift for parts (for example, if I receive clothes I would never wear, I might up-cycle or re-purpose them). This is just how it goes if you have limited space, selective taste, or are particular about contents/message of a toy, etc.