Unit testing void methods?

If a method doesn't return anything, it's either one of the following

• imperative - You're either asking the object to do something to itself.. e.g change state (without expecting any confirmation.. its assumed that it will be done)
• informational - just notifying someone that something happened (without expecting action or response) respectively.

Imperative methods - you can verify if the task was actually performed. Verify if state change actually took place. e.g.

void DeductFromBalance( dAmount ) 

can be tested by verifying if the balance post this message is indeed less than the initial value by dAmount

Informational methods - are rare as a member of the public interface of the object... hence not normally unit-tested. However if you must, You can verify if the handling to be done on a notification takes place. e.g.

void OnAccountDebit( dAmount )  // emails account holder with info

can be tested by verifying if the email is being sent

Post more details about your actual method and people will be able to answer better.
Update: Your method is doing 2 things. I'd actually split it into two methods that can now be independently tested.

string[] ExamineLogFileForX( string sFileName );
void InsertStringsIntoDatabase( string[] );

String[] can be easily verified by providing the first method with a dummy file and expected strings. The second one is slightly tricky.. you can either use a Mock (google or search stackoverflow on mocking frameworks) to mimic the DB or hit the actual DB and verify if the strings were inserted in the right location. Check this thread for some good books... I'd recomment Pragmatic Unit Testing if you're in a crunch.
In the code it would be used like

InsertStringsIntoDatabase( ExamineLogFileForX( "c:\OMG.log" ) );

As always: test what the method is supposed to do!

Should it change global state (uuh, code smell!) somewhere?

Should it call into an interface?

Should it throw an exception when called with the wrong parameters?

Should it throw no exception when called with the right parameters?

Should it ...?

Test its side-effects. This includes:

• Does it throw any exceptions? (If it should, check that it does. If it shouldn't, try some corner cases which might if you're not careful - null arguments being the most obvious thing.)
• Does it play nicely with its parameters? (If they're mutable, does it mutate them when it shouldn't and vice versa?)
• Does it have the right effect on the state of the object/type you're calling it on?

Of course, there's a limit to how much you can test. You generally can't test with every possible input, for example. Test pragmatically - enough to give you confidence that your code is designed appropriately and implemented correctly, and enough to act as supplemental documentation for what a caller might expect.