The answer would vary depending on which period of Athenian history we're talking about, but I'm assuming that you're asking about the Athenian democracy (roughly 507-322 BC). In which case:

My understanding is that only citizens of Athens could vote--and that a citizen had to be a landowning man.

The first part is right, the second isn't. The fundamental principle of the democracy was that all male Athenian citizens could vote (and speak in the Assembly, and sit on the Council, and serve in the jury courts). It didn't matter whether they were wealthy estate owners or penniless beggars. There was no property requirement for political rights.

A property requirement for full citizenship was regarded as a typical feature of oligarchy; the lack of such a requirement was one of the defining traits of Greek democracy. Indeed, while Athens technically maintained property requirements for certain high offices, we're told that these were not really enforced, and that poor citizens could get into high office by lying about their wealth. After all, anyone caught trying to bar poor people from having a voice in politics could too easily be painted as an oligarch and an enemy of democracy, which would have very serious consequences.

So what did it take to become a citizen? After the introduction of Perikles' Citizenship Law of 451 BC, the requirements were pretty straightforward: both your mother and your father had to be Athenian citizens. If they were, then you were born an Athenian citizen, and all you had to do was make sure you were registered on the deme rolls when you turned 18. If they were not, you could not become an Athenian citizen except by vote of the Assembly. They very rarely awarded citizenship to particularly generous or loyal individuals from other states; they also, in a few cases, offered blanket Athenian citizenship to all citizens of a particular allied state. But these were exceptions, and citizenship status was normally jealously guarded. Every now and then there would be an infiltration scare and citizens would be required to prove their ancestral right to their citizen status; many citizens faced court cases over accusations that they had illegitimately claimed full citizenship for themselves or their children.

In fact, if there had been a requirement for citizens to own land, it would have been another impossible hurdle for non-citizens, because they were not allowed to buy land or real estate. Only citizens could own land within Athenian territory. This was a way to keep property exclusively in citizen hands, but also to prevent foreign families from establishing a permanent presence and eventually insinuating themselves into their deme registry.

In short, there was no land ownership requirement for citizenship in democratic Athens; the lack of such a requirement was a key feature of its democratic system. On the other hand, citizenship was an extremely closely guarded privilege of native Athenians (male and female) and only Athenian men could vote.