: : : : And yes, I do know that Middle-earth is an IMAGINARY place, but still...

: : : The population of Middle Earth ranged from none, prior to the awakening of the elves, to a whole bunch at the end of the Third Age. Seriously, there is really no way to accurately or questimate a population for Middle Earth after the awakening of the elves. ~Aelmer

: : Are you counting Dragons, Giants, Great Eagles, Wargs, Great Spiders, Ravens, Crows, Thrushes, Huorns, Foxes, and all the other inteligent yet non-Free People non-Enemies beings in Middle-earth.

: I believe the question was about the population of peoples ( Elf, Man, Dwarf ). I suppose you could include orcs, but there is still now way to determine what the population of ME was at any given time, except for time prior to the awakening of the elves. At that time, the number of peoples inhabiting ME would be zero.

True, but I have seen some impressive estimates done - especially in 'Other Hands' a magazine devoted to game playing in Tolkeins world - that gave an estimate for the population of Gondor from ca 11 million (at its height) to 5 million (at the time of the war of the ring).
We know quite a bit about individual sites - Bywater/Hobbiton ('every day was someone's birthday') so a minimum population of 365 and probably more like 1000. You could check the spread of towns in the Shire - with this as a standard population for Shire towns, and then as it is heavily agriculturally based, extrapolate a rural population of prob 3 times the towns based in hamlets and farms/villages and work out a rough figure - I would say a quarter of a million at the time of the War of the Ring.
Bree has 'a hundred stone houses of the big folk' plus hobbit holes. Even at four to a house plus hobbits thats 600, again its probably near 1000. Lake Town has been discussed elsewhere but prob has a population of 1500 or so.
'Other Hands' used a popluation per square mile base, using the type of realm (Classical, Medieval, stone age) etc and multiplied the area in sq miles by the historical evidence from our world for population density in such places, not a bad way of guessing should you so wish.
As a toal guesstimate, I would say between 50 and 100 million sentient beings in Middle-Earth all told at the end of the Third Age, more towards 100 million, but thats guess work.
Guesstimates I would say are (at WotR)
Rohan - 400,000
Dunland - 300,000
Eriador incl. Shire - 500,000
Gondor - 5,000,000
Lindon/Grey Havens - 20,000
Rivendell - 1,000
Anduin Vales (northmen/beornings) - 50,000
Woodmen - 20,000
Mirkwood Elves - 50,000
Lorien - 25,000
Dale/Esgaroth - 250,000
Dwarves (Durins Tribe) - 100,000
Umbar(and territories) - 500,000 to 1,000,000
Mordor (and Nurn/Khand etc.) - 2 or 3 Million
With the rest spread out over Middle Earth - based around river valleys, mountain vales, and steppes.
Don Quixote