You're basically right about data organization. If you have cases organized like this:

ID M1 M2 M3 EVENT
You will likely want to reorganize the data so that it looks like this:

ID TIME EVENT
1  1    0
1  2    1
1  3    1
2  1    0
2  2    0
.  .    .
.  .    .
I call this a conversion from a wide format to a long format. It is done easily in R using the reshape() function or even more easily with the reshape2 package.

I personally would keep the ID field for its potential use in identifying a source of variation in a mixed effects model. But this is not necessary (as pointed out by @BerndWeiss). The following assumes you would want to do so. If not, fit a similar model with glm(...,family=binomial) without the random effect terms.

The lme4 package in R will fit a mixed effects logistic regression model similar to the one you're talking about, except with a random effect or two to account for variability in the coefficients across subjects (ID). The following would be example code for fitting an example model if your data are stored in a data frame called df.

require(lme4)
ans <- glmer(EVENT ~ TIME + (1+TIME|ID), data=df, family=binomial)
This particular model allows the TIME and the intercept coefficients to vary randomly across ID. In other words, this is a hierarchical linear mixed model of measurements nested in individuals.

An alternate form of a discrete time event history model breaks TIME into discrete dummies and fits each as a parameter. This is essentially the discrete case of the Cox PH model because the hazard curve is not restricted to being linear (or quadratic, or however you can imagine transforming time). Although, you may wish to group TIME into a manageable set (i.e. small) of discrete time periods if there are a lot of them.

Further alternates involve transforming time to get your hazard curve right. The previous method basically alleviates you from having to do this, but the previous method is less parsimonious than this (and the original linear case I posed) because you may have a lot of time points and thus, a lot of nuisance parameters.

An excellent reference on this topic is Judith Singer's and John Willet's Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence.

Singer and Willett have been published a lot on this subject. I highly recommend that you read some of their papers. You also might want to get their book "Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence". Clearly one of the best textbooks in this field.

For most book chapters there is R sample code (see chapters 11ff) available that demonstrates how your data has to be structured ("person-period format") and how to analyze that kind of data. For a standard discrete-time model you do not need the ID variable and you also do not need to estimate a mixed-effects model as suggested by @ndoogan. A simple glm(event ~ time + ..., family = "binomial") works just fine. Singer and Willett also discuss many issues how to model the time variable (linear, quadratic, ...)

To cite two more references that I highly recommend:

Allison (1982): "Discrete-Time Methods for the Analysis of Event Histories" (PDF) (the Allison article also discusses why you can use a standard glm instead of a mixed-effects model)
Mills (2011): "Introducing Survival and Event History Analysis"

Bernd Weiss