Investors who own higher-yielding equities do so for the yield. If they can get a similar yield with a government bond they will invest in the government bond and not the company. All things being equal, if government bond yields go up, then yields from dividend-producing companies also have to go up for investors to be interested in them, and the only way that can happen is if their stock price goes down.
Equities with low divvies or no divvy are dependent on growth to keep investors happy. A company with no growth and no divvy is stagnant and will be bid down to its book value or worse.
Companies which are highly-leveraged (e.g. [m]REITs) have much higher borrowing costs than companies which are not and these borrowing costs DIRECTLY effect their dividend. Thus if interest rates go up, it has a DIRECT effect on these companies.. their yields will go down (due to having to reduce their divvies) at the same time bond yields are going up. A DOUBLE whammy, so to speak. Their stock prices will crash.
Companies which are highly-indebted (e.g. MLPs) have higher costs for future growth at the very least, but higher interest rates do not immediately effect their current distributions. This isn't quite a double whammy but it's pretty bad.
Companies which have very low debt are almost immune to rises in interest rates. These companies will be able to keep pace with inflation + grow on top of that regardless of what interest rates are. Remember that interest rates != inflation. This is why you see the market always reverting to a nominal positive inflation-adjusted growth rate in the long-term ~80-100 year market graphs. Inflation can push down stock prices in the near term shock but ultimately doesn't have much of an effect. It's interest rates that have the larger shock and larger long-term effect.