Linkable ring signatures have found many attractive applications. One of the recent important extensions is a linkable threshold ring signature (LTRS) scheme. Unfortunately, the existing LTRS schemes are only secure in the random oracle model (ROM). In this paper, we make the following contributions. First, we construct the first LTRS scheme that is secure without requiring the ROM. Further, we enhance the security of a threshold ring signature (for both linkable or non-linkable) by providing a stronger definition of anonymity. This strengthened notion makes threshold ring signature schemes more suitable in real life. Finally, we provide efficient schemes that outperform the existing schemes in the literature. Our scheme is particularly suitable for electronic commerce or electronic government where anonymity and accountability are the most concerned factors.