From cfrick's suggestion, I tried this, (applies to grails 3)
Create a filter e.g. [grails-project]/src/main/java/hello/SimpleCORSFilter.java
(Create the 'java' directory, or create a SimpleCORSFilter.groovy in the [grails-project/src/main/groovy directory)
You can use the example in one of the spring guides.
package hello;
import java.io.IOException;
import javax.servlet.Filter;
import javax.servlet.FilterChain;
import javax.servlet.FilterConfig;
import javax.servlet.ServletException;
import javax.servlet.ServletRequest;
import javax.servlet.ServletResponse;
import javax.servlet.http.HttpServletResponse;
import org.springframework.stereotype.Component;
@Component
public class SimpleCORSFilter implements Filter {
    public void doFilter(ServletRequest req, ServletResponse res, FilterChain chain) throws IOException, ServletException {
        HttpServletResponse response = (HttpServletResponse) res;
        response.setHeader("Access-Control-Allow-Origin", "*");
        response.setHeader("Access-Control-Allow-Methods", "POST, GET, OPTIONS, DELETE");
        response.setHeader("Access-Control-Max-Age", "3600");
//        response.setHeader("Access-Control-Allow-Headers", "x-requested-with");
        // I used this instead
        response.setHeader("Access-Control-Allow-Headers", "Origin, X-Requested-With, Content-Type, Accept");
        chain.doFilter(req, res);
}
    public void init(FilterConfig filterConfig) {}
    public void destroy() {}
}
Then in [grails-project]/grails-app/conf/spring/resources.groovy, add the filter, e.g.
import hello.SimpleCORSFilter
// Place your Spring DSL code here
beans = {
    myFilter(SimpleCORSFilter)
}
Also, in the grails docs, to order your filter:
import hello.SimpleCORSFilter
import org.springframework.boot.context.embedded.FilterRegistrationBean
import org.springframework.core.Ordered
myFilter(FilterRegistrationBean) {
    filter = bean(SimpleCORSFilter)
    urlPatterns = ['/*']
    order = Ordered.HIGHEST_PRECEDENCE
}