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: 
Yes, you are making a mistake in the theory you know. Each major key sig. has a relative minor to go with it - C- Am: G- Em; Bb- Gm etc. Whilst the majors adhere exactly to the # or b in their key sig., minors cannot and do not.

That's because the notes that make up the minor scales vary in their 6th and 7th notes. It's the way they are. Take A minor. 6th note can be F or F#, 7th can be G or G#. In your scenario, E minor can have C or C#, and D or D#.

The D/D# anomaly makes the chord based on B either B minor or B major respectively. Both are quite acceptable, and both are used. The B major, as you find by using it and it sounds good to you, uses the leading note of D# to sound more decisive going into E minor, but there's nothing in the theory that says we must use that, or we must use B minor. Whatever fits better is what we should use. Please don't use theory as a set of *rules*. It's more a set of *guidelines*. Usual adage - if it sounds good, it usually is - whether the 'rules' get broken or not!
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.