This rosetta stone, some don't like the word cheat sheet has helped me in the past.
Match the melody note with a chord that has that melody note in it's makeup. For example:
Degree of the scale - think of using one of these chords.
1 try the I, IV, vi or ii7 For example C note, then C chord (CEG), F (FAC), Am (ACE)or Dm7 (DFAC)would harmonize.
2 (D) try the V (GBD), ii7 (DFAC) or iii7 (EGBD)
3 try the I, vi or iii.
4 try the IV, ii or V7
5 try the V, I or iii
6 try the IV, ii or vi
7 try the V7 or iii.
Relying on this you could have a chord for every melody note - that is not a good idea. When the melody moves away from notes used in the old chord your ear will alert you something needs to change, i.e. it is time to find a chord that will fit (harmonize)with the new melody notes.
This little chart has helped me flesh out a progression. Like Yoke said the I IV V is a safe place to start then move to the minor chords in the key --- one of them will harmonize.

As to just one left hand note[/b], looking at this chart the Root seems to do this, i.e. C melody, C chord harmonizes so just use root C. D melody, Dm chord harmonizes again use D root, E melody, Em chord use E. The B did not follow this logic, however diminished seldom do. Not near a keyboard, but, I think I'm safe here.
{Edit} Would't this just be a hugh octive