There are 2 non-plastic cutting board options that I'm aware of that are dishwasher safe and aren't damaging to knives like glass boards.
First, there are the rubber cutting boards popular with commercial kitchens, sold under brand names like Sani-TUFF. While some sources (e.g. KaTom Restaurant Supply) claim these have to be hand washed, they are popular in commercial kitchens because they can handle the heat of the commercial sanitizing dishwashers without warping or melting, and they don't pick up scratches as easily as plastic. They are also generally anti-microbial, I assume because they are non-porous. Although these are usually synthetic rubber, they are much harder than plastic cutting boards -- though not too hard on your knives -- and the likelihood of you consuming any part of them inadvertently should be pretty small.
Next, there are wood fiber composite cutting boards. The brand I know best of these is the Epicurean label. They are made from various specific materials, but the Epicurean label is made from a paper composite called Richlite. Assuming it's the blend discussed on the Richlite page, it is 65% recycled paper and 35% phenolic resin. While that is a synthetic polymer, it's much harder than your typical cutting board and is unlikely to melt (up to 350°F) or react with anything it's likely to come into contact with in your kitchen. These boards are also quite dense, so again, the likelihood of you consuming any of it unintentionally is exceedingly low.
Both of these products are also "workable", meaning you could cut them to custom sizes or shapes and sand them if they ever get scratched up enough to need it.
For what it's worth, environmental sources of microplastics are a much stronger vector than anything in your kitchen, and many of these are practically unavoidable. However, if you are concerned about plastic from cutting boards, both of the above options are far more durable and less likely to introduce plastic into your food but are more convenient than wood in terms of cleaning. Of course, I encourage you to research these on your own and use your best judgment.