Alternating Ovulation
Nope, not true.  It's a pretty common myth; that paper cites a  reference from 1932 in primates claiming to show that the ovaries alternate.  A few papers before this, in particular this one from 1991 showed that ovulation was not alternating, but this was the first to do so in actually fertile woman.  In particular,


  Alternate ovulation occurred in 61 of the 119 succeeding ovulations (51.3%, not significant) which means that there was no consistent tendency to alternation. 


Alternation only occurred half the time.  Neither did they find a difference between side preference; right-side ovulation occurred in just over 50% of the cases, not significantly different.

FSH, follicle-stimulating hormone, and LH, luteinizing hormone, stimulate each ovary equally, but one ovary will get slightly ahead.  Ovulation (and indeed pregnancy at times) is a positive-feedback loop, meaning the response will be exponential; if one ovary is even just slightly ahead it will rapidly dominate the other.  The less-developed one quickly senses the other is ahead of the game and ceases to respond.