Solution 1:

New System.Text.Json serialization

ASP.NET MVC Core 3.0 uses built-in JSON serialization. Use System.Text.Json.Serialization.JsonStringEnumConverter (with "Json" prefix):

services
    .AddMvc()
    // Or .AddControllers(...)
    .AddJsonOptions(opts =>
    {
        var enumConverter = new JsonStringEnumConverter();
        opts.JsonSerializerOptions.Converters.Add(enumConverter);
    })

More info here. The documentation can be found here.

If you prefer Newtonsoft.Json

You can also use "traditional" Newtonsoft.Json serialization:

Install-Package Microsoft.AspNetCore.Mvc.NewtonsoftJson

And then:

services
    .AddControllers()
    .AddNewtonsoftJson(opts => opts.Converters.Add(new StringEnumConverter()));

Solution 2:

some addition:
if use Newtonsoft.Json

Install-Package Microsoft.AspNetCore.Mvc.NewtonsoftJson
services
    .AddControllers()
    .AddNewtonsoftJson(options =>
        options.SerializerSettings.Converters.Add(new Newtonsoft.Json.Converters.StringEnumConverter()));

options.SerializerSettings.Converters

SerializerSettings is necessary