AMT for most W-2 earners is just a timing difference (meaning you'll end up "using up" the AMT excess at some point anyway). From a cashflow perspective, I get wanting to avoid it, but if you're fairly sure the ISOs won't turn up worthless, I recommend taking a look at the capital gains prospects instead.
Suppose you think the company will IPO (or be acquired by a public company) in N years and you'd like to sell the last shares at N + P years. (P = 1/2 if you want to sell immediately after a 180 day typical lockup period.)
Take (N + P + 1) and divide the number of ISOs you have by that figure.
Figure out any oddity around tax year boundaries.
Buy (exercise and hold) that many per year.
This ensures that you will buy the least amount per year [minimizing cashflow and AMT] and for all shares sold you will meet the 2 years since grant and 1 year holding period required to have the gain treated as a long-term capital gain. If there are other tax incentive programs that the stock qualifies for (typically not for an IPO-likely company), make sure you meet those terms and holding periods if desired.
Alternately, if you think there's still risk of going to $0 or an IPO being delayed in a market meltdown, do the math above, but calculate the "N" as starting 2 or 4 years from now and implement this plan 2 or 4 years from now (more shares per year, but zero risk in the interim).
AMT is not necessarily something to be avoided at the expense of other cashflow, risk, and capital gains term questions. I got whacked a lot by AMT in 1996, but was able to use up all the credits in only a few years after that. I wish I'd have better managed the short vs long term capital gains aspects, even at the expense of paying more AMT temporarily.