Take one cube (the "first cube"), you can colour this in 3 different ways.

Suppose we colour this first cube red, then there are 3 different choices for the second cube, and 3 for the last cube. This gives a total of 9 choices, with the first cube red. Now some of these coincide (by "permuting" the cubes - which means "moving them around"), for example R,B,G and R,G,B are the same. The choices are (consider a "tree like diagram" - if you want me to ellaborate on this i will) RRR, RBR, RBB, RGG,RGR and RGB... 6 in total.

Now if we choose our first cube as blue, then our second cube can take on 3 different colours. But the colour of this second cube cannot be red, because then the final cubes colour will (by permuting the cubes) give a colour that was already done in the above case (where we choose red as the colour of the first cube). Thus we have two choices for the colour of the second cube: green or blue. Now the colour of the last cube can also take on two colours only (namely blue or green), since chosing the colour red (by permuting the cubes) will give a choice of colours already given in the stage where we chose red as the colour of the first cube. Thus, for blue as the colour of the first cube, we have 2*2=4 different choices (Green and Blue, Blue and Green, Blue and Blue and Green and Green). This gives us a total of 3 choices for the 3 cubes colours (with blue as the colour of the first cube): BGB BGG, BBB.

Finally lets let the first cube have colour green. We cannot choose the second cube to have colour red (because, by permuting, this combination will, on chosing a colour for the last cube, be a colour covered in the first paragraph), and similarly we can't let this second cube have colour blue (because this will, by permuting, lead to a case covered in the second paragraph). Thus the second cube must have colour green. Similarly the third cube must have colour green. Thus, with green as the colour of the first cube, we can only have 1 choice for the colours of the other cubes - Green and Green. This gives the option GGG.

Therefore there are: 6 + 3 + 1 = 10 choices for colourings.

[I'm going to read over this and check that I haven't made a mistake with cases, but perhaps people want to read this and give me some feedback]

Hope this helps! If not I hope it points you in the right direction...