I would add to what Bryan already said that while keeping URL structure is important, or at least doing proper 301 redirects if keeping URLs is not possible, a redesign is an opportunity to improve your rankings (but also to lose them) by changing a number of on-site elements.
Keep in mind at least the following:
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unique titles
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unique meta descriptions
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proper use of h1 / h2 / h3 headers
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proper use of your target keywords in anchor text for internal links
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proper use of , and list elements
**_- proper use of alt and title tags for images
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please consider using microformats (schema.org) if they make sense in your case
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proper internal linking (analyze if all your internal pages are easily reachable. Even without changing URLs, you can still add or remove links on individual pages to improve the flow)
These are all elements which are easily changed during a redesign - for better or for worse.
Hope it helps in your decisions._**