Mayonnaise Substitutes
MiracleWhip, of course...

If it's just the taste of the oil that bothers you, you can easily make your own mayo using whatever oil you wish. A cheap extra-virgin olive oil makes a delicious mayonnaise, IMHO. 

Otherwise, use vinegar. It's a great general-purpose condiment, can be flavored easily, and there are plenty of varieties to choose from. Naturally, this won't work well if you need the body of mayo, so go with some form of prepared mustard in those cases - it's like emulsified vinegar (in that it's emulsified vinegar).