Clearly, you care at least a little bit about what others think, ot you wouldn't be saying "I wonder if people will think he's being cheap." If you didn't care, you wouldn't even think about it.
The bottom line is this - an engagement ring is a symbol of a promise - nothing more, nothing less. That symbol can be whatever you want it to be. If you'd prefer an engagement ring with a gemstone, with a fake stone or with no stone, the symbol is the same. Heck, if you want an engagement necklace instead, that's up to you and your boyfriend. Women probably will say something (perhaps not directly to you) if you don't have the traditional diamond engagement ring. Women talk about other women, and most women talk about other women's engagement rings. If you had a huge diamond, they'd talk; if you had a tiny diamond, they'd talk; and if you have no diamond, they'll talk. You can't let that play a role in your decision.
The symbol of your engagement is about you and your boyfriend. Period. Tradition and other people's opinions shouldn't play a role in this.