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checking and raising is a more expensive bluff, and betting can be cheaper.

betting also give you better implied odds on average, and 3 bets go in less often.

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Betting out can also be expensive since this line increases the probability that this will still be a 3 player confrontation on the turn, which means the hero will be significantly less likely to take this pot down unimproved.

About the implied odds argument. With a strong draw such as a flush draw in a 3 handed raised pot, I think its better if the hero takes the line that is most likely to win this pot as soon as possible, rather than try to draw cheaply. Just telling you how I approach this situation, not saying your way is wrong and my way is right.