下面是一个按照月份名称非字母顺序排序的LINQ查询的解决方法的代码示例：

using System;
using System.Linq;

public class Program
{
    public static void Main()
    {
        string[] months = { "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" };

        var sortedMonths = months.OrderBy(m => GetMonthOrder(m));

        foreach (var month in sortedMonths)
        {
            Console.WriteLine(month);
        }
    }

    public static int GetMonthOrder(string month)
    {
        switch (month)
        {
            case "January":
                return 1;
            case "February":
                return 2;
            case "March":
                return 3;
            case "April":
                return 4;
            case "May":
                return 5;
            case "June":
                return 6;
            case "July":
                return 7;
            case "August":
                return 8;
            case "September":
                return 9;
            case "October":
                return 10;
            case "November":
                return 11;
            case "December":
                return 12;
            default:
                throw new ArgumentException("Invalid month name");
        }
    }
}

这段代码首先使用OrderBy方法对months数组进行排序，排序的依据是GetMonthOrder方法返回的月份顺序数字。GetMonthOrder方法根据月份名称返回对应的顺序数字。最后，使用foreach循环遍历排序后的月份数组，并打印每个月份的名称。