There are basically three different approaches.

1. An approach based on cubic forms, which was used by Karim Belabas to quickly list cubic fields with discriminant up to a certain bound.

2. The approach via class field theory mentioned above: look at all quadratic number fields with small discriminant, find all of its cubic cyclic extensions via class field theory, and check which of these are not abelian over the rationals (this set includes all whose conductor is not invariant under the Galois group of the quadratic field).

3. The approach via Kummer theory: start as in 2., but then adjoin a cube root of unity; cyclic cubic extensions will simply be Kummer extensions over this extension. Standard methods (this should be in Gras' book on class field theory) allow you to step down once the construction is done.

Approaches 2 and 3 go back to Hasse [Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage; Math. Z. 31 (1930), 565-582] and Reichardt [Arithmetische Theorie der kubischen Körper als Radikalkörper; Monatsh. Math. Phys. 40 (1933), 323-350].

There are basically three different approaches.

1. An approach based on cubic forms, which was used by Karim Belabas to quickly list cubic fields with discriminant up to a certain bound.

2. The approach via class field theory mentioned above: look at all quadratic number fields with small discriminant, find all of its cubic cyclic extensions via class field theory, and check which of these are not abelian over the rationals (this set includes all whose conductor is not invariant under the Galois group of the quadratic field).

3. The approach via Kummer theory: start as in 2., but then adjoin a cube root of unity; cyclic cubic extensions will simply be Kummer extensions over this extension. Standard methods (this should be in Gras' book on class field theory) allow you to step down once the construction is done.

Approaches 2 and 3 go back to Hasse [Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage; Math. Z. 31 (1930), 565-582] and Reichardt [Arithmetische Theorie der kubischen Körper als Radikalkörper; Monatsh. Math. Phys. 40 (1933), 323-350].

Edit: I also should have given the origin of approach 1, since it is most often (incorrectly) credited to Delone and Faddeev. In the preface to their book Irrationalities of the third degree, however, they do credit F.W. Levi with this observation: see Kubische Zahlkörper und binäre kubische Formenklassen (Cubic number fields and classes of binary cubic forms), Leipz. Ber. 66 (1914), 26-37