With players that you definitely want to keep, when do you guys offer your long-term contracts, 2nd year arb or 3rd year arb?  Have you guys seen a lot of downside with waiting until their 3rd year?

If I'm not misreading your post, you're asking 2 arbs and a long-term deal.    If that's the case, the LT deal is usually more expensive but I've never had a player refuse to sign after 2 arbs.   It's after the 3rd arb when the problem comes.

From what I remember about my recent arb cases, the cost of a long term and an arb signing for the 2nd arb tend to be about the same.  Because the major benefit of arb signings is a cheaper-than-market salary, I tend to offer the long term contract when I'm not getting an arb discount, so that tends to be the 2nd arb.  Granted, that means one less year of gauranteed time on the club.

Of course, if the guy has a low health rating, I tend to arb him longer in case he catches a major injury

Sorry, it looks like I was a little unclear.  I should have said, "when they're up for 2nd arb vs 3rd arb".  I've already been burned by waiting until after 3rd arb to go for long-term.  I was just wondering if anyone had been burned by waiting until they're up for 3rd arb (already did 2 years of arb) before offering that long-term contract.  From MikeT's response, it sounds like that's still safe.

For the first two ARB years I usually do whatever is better (i.e. cheaper) from a financial standpoint, which is usually offering ARB.  If he's a player I want to keep, I'll offer a LT deal in the 3rd year.  I've never seen a player turn down a LT deal while still in his ARB years.