How do I figure out the relative polarity of organic compounds?
General steps to figure out the polarity (or at least relative polarity) and subsequently the relative boiling point of compounds:


Acquire the correct geometry
Estimate dipole moments
Sum up estimated dipole moments
Check for other intermolecular interactions


Acquire Correct Geometry

This step requires some knowledge about the geometry of hydrocarbon compounds. But it seems that you already know about the tetrahedral bonding centers of saturated hydrocarbons. This is then important later on for the dipole forces. Never try to linearize the carbon backbone just to try to make sense of this, it will do more damage than good.

Estimate Dipole Moments

The dipole moments that you have to estimate will be those along bonds. There is a larger dipole moment in the ether than the hydrocarbon without heteroatoms, because the oxygen atom is strongly electronegative. As such, there will be a dipole along the $\ce{C-O}$ bond. 

Sum Up Estimated Dipole Moments

These dipoles can be understood as vectors along bonds, pointing from positive to negative partial charges, and can be summed up for the total dipole moment of a molecule. The larger the absolute value of this dipole vector is, the larger the dipole-dipole interactions between molecules: Subsequently, the higher the forces between the molecules of this substance.

Check for Other Intermolecular Interactions

One other main intermolecular action is, as you already pointed out in your question, hydrogen bonding. This one is quite strong, but isn't found for every compound out there. Basically, it is when a highly polarized $\ce{H-X}$ bond can be formed, where $\ce{X}$ can be oxygen, nitrogen, chlorine or some other electronegative atom (with respect to carbon). The hydrogen that is sandwiched in the hydrogen bond has to be (formally) bonded to such an electronegative atom as well, otherwise it wouldn't be as free to create those bonds (as you also have realized, because the hydrocarbon hydrogens are really quite boring from an intermolecular interaction perspective).

Result

Based on the steps above, we can generate a table:



The more "Yes" a substance has, the higher the intermolecular forces. Based on these data, assumed by us, we can generate a list (from highest boiling point to lowest):


Ethanol
Dimethyl ether
Propane