CSS div alternating colour

Don't use nth-child, use nth-of-type

div.container > div:nth-of-type(odd) {
    background: #e0e0e0;
}

.container {
  width: 600px;
  margin: 0 auto;
}

.row {
  line-height: 24pt;
  border: solid 1px black;
}

div.container>div:nth-of-type(odd) {
  background: #e0e0e0;
}

h3 {
  line-height: 36pt;
  font-weight: bold;
  color: blue;
}

<div class="container">
  <h3>Title</h3>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <h3>Title</h3>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <h3>Title</h3>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <h3>Title</h3>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
  <div class="row">Content</div>
</div>

You probably want to match on type, not child.

Use :nth-of-type such as

.row:nth-of-type(odd) {
    background: #e0e0e0;
}

The easiest solution is of course just to wrap the elements you want striped.

Your updated jsFiddle.

HTML

<div class="container">
    <h3>Title</h3>
    <div class="zebra">
        <div class="row">Content</div>
        <div class="row">Content</div>
    </div>
    <h3>Title</h3>
    <div class="zebra">
        <div class="row">Content</div>
        <div class="row">Content</div>
        <div class="row">Content</div>
    </div>
    <h3>Title</h3>
    <div class="zebra">
        <div class="row">Content</div>
        <div class="row">Content</div>
        <div class="row">Content</div>
        <div class="row">Content</div>
    </div>
</div>

CSS

.row:nth-child(odd) {background: #e0e0e0;}

Bear in mind also that if browser support is important to you, you might want to generate additional classes for zebra-striping server side instead.