Question: I find when playing in a minor key I always end up playing the major version of the V chord. (So in Em I always end up playing a "B major" chord.)

However, the third of this chord should be played as a minor when looking at the key and it seems like using an accidental as frequently as I do is not really in the spirit of accidentals.

Am I making a mistake in my music theory? Am I playing the key I think I am? Is this just a mode? Do I need to start playing more B minors?

Please bear in mind I am relatively new to music theory and would appreciate it if you treat me like an idiot. Thanks.

Answer: 
Although the D in the scale of B will be natural in the key signature, it will often be sharpened when leading to the key note E. Think of the melodic minor scale for E. D will be sharpened on the way up but natural on the way down. 

So, if your chord is to resolve to the tonic E, the sharpened D will work nicely. You might find that a B minor chord works well if it goes somewhere else next.
One way to look at things is that, for music in a minor key, the chord on scale step 5 is minor (Gm in the key of Cm). In a cadence, scale step is usually raised to be a half step below the tonic (scale step 8). In non-cadential passages, often the minor chord on the 5th step is used. For example, in a cycle of fifths, one might have (in the major): I,IV,vii0,iii,vi,ii,V,I etc. In a minor key, an ending cycle might look like: i,iv,VII,III,VI,ii0,V,i. However if the passage contains two cycles one often gets: i,iv,VII,III,VI,ii0,v,i,VII,III,VI,ii0,V,i. The "interior" part of the passage is just a string of chords, not a cadence. Of course, this treatment is optional and neither is incorrect; they are just different.

A similar example occurs in sequence decent by thirds (Pachelbel canon for one example). In a major key: I,V,vi,iii,IV,I6,ii6,V (where the descending bass turns around for the last two chords, not the only choice.) In minor keys the following is often found, i,v,VI,III,iv,i6,ii06,V in a cadential context; the last chord can be v if not meant cadentially. 

As an aside: this sequence is very flexible. In a minor key, the chords on the scale step 5 can be either major or minor; the iii chord can be a I64 (not really cadential or arpeggiated); the ii0 chord can be iv; the ii chord can be IV; the ii or ii0 chords can be in root position; and lastly the final V (or v) chord can be a vii06 leading to a I (or i) chord which can give a bass line that descends an octave.