Bryan Anderson in the comments below should get the credit for this, so feel free to repost Bryan so I can give you the points.
You have to do a manual build of the LWC to get this feature, at least with the knowledge I have, that is what it took.
- https://www.lightningdesignsystem.com/components/cards/
Copy this code:
<template>
<article class="slds-card">
<div class="slds-card__header slds-grid">
<header class="slds-media slds-media_center slds-has-flexi-truncate">
<div class="slds-media__figure">
<span class="slds-icon_container slds-icon-standard-account" title="account">
<svg class="slds-icon slds-icon_small" aria-hidden="true">
<use xlink:href="/assets/icons/standard-sprite/svg/symbols.svg#account"></use>
</svg>
<span class="slds-assistive-text">account</span>
</span>
</div>
<div class="slds-media__body">
<h2 class="slds-card__header-title">
<a href="javascript:void(0);" class="slds-card__header-link slds-truncate" title="Accounts">
<span>Accounts</span>
</a>
</h2>
</div>
<div class="slds-no-flex">
<button class="slds-button slds-button_neutral">New</button>
</div>
</header>
</div>
<div class="slds-card__body slds-card__body_inner">Anything can go into the card body</div>
<footer class="slds-card__footer">
<a class="slds-card__footer-action" href="javascript:void(0);">View All
<span class="slds-assistive-text">Accounts</span>
</a>
</footer>
</article>
</template>
Then mark the 'slds-card' with the background selector and change the color in a CSS file or Inline-Style (didn't try inline but assuming it will work).
I don't like this solution but it appears to be the only way to get it to work.
article
element and then expose it as an attribute to modify depending upon your use case