Word-initial [h-] in Spanish was gradually lost in most peninsular dialects by the late Middle Ages, but this sound change was not universal, and this sound is still present in colloquial Andalusian, Extremaduran, and Canarian dialects today:^{4 5 6}

The loss of [h-] did not spread everywhere, of course, since it was a relatively late innovation. Nebrija, for example, at the end of the fifteenth century, described [h] as a regular part of the phonemic inventory of Spanish. Today [h-] is still found in popular speech in Extremadura, Huelva, Sevilla, Cádiz, Córdoba, Málaga and the western portions of Granada and Almeria (Penny 1972-3, 470), as well as in many parts of Spanish America.

• From Latin to Spanish: Historical phonology and morphology of the Spanish language (p.325)

As well as in various cosmopolitan dialects descended from colonisers from the above regions (e.g. Mexico¹, the Philippines², Equatorial Guinea³).

This aspiration has even lead its way into the orthography of some regionalisms, replacing "h-" with "j-":

Esta aspiración aún se conserva como rasgo dialectal en Andalucía, Extremadura, Canarias y otras zonas de España y América. A veces, la aspiración llega casi a convertirse en el sonido velar fricativo sordo /j/, pronunciación que en algún caso tiene reflejo en la escritura; así ha ocurrido, por ejemplo, con el adjetivo jondo (‘hondo’, del lat. fundus), que se aplica al cante más genuinamente andaluz, caracterizado por su profundo sentimiento, o con el verbo jalar, variante de halar usada en varios países americanos, o con jolgorio, grafía hoy mayoritaria frente a la etimológica holgorio.

• DPD: h (2.)

<sup>1. A Panorama of Traditional New Mexican Spanish (p.8-9)
2. A History of the Spanish Language (1.5.4 The Philippines)
3. "Cuba Africana" (1858-1898), transferencias lingüísticas afrocubanas (p.46)</sup>

<sup>4. Variation and Change in Spanish (4.1.7.2.3 Maintanence and loss of /h/, p.121-122)
5. Andalusian Spanish (2.3 Retention of [h] < Latin f)
6. DPD: h (2.)</sup>

The H in hacienda (or really anywhere else¹) is not pronounced at all. It's a silent letter.

The X used to have a sound similar to the SH in English and the J had a sound similar to English's J (if you're familiar with the sound often written as ZH, that's it). Over time, the sounds represented by the X and the J, which were fairly close, merged into a single sound that evolved into something that sounds quite like the English H.

Because many Xs now sounded the same as Js, most words that had an X were changed to J (those Xs that sounded like English's X kept their X as a rule).

So why does X get used for the H sound today? Many places in Mexico got their name before the sound change happened (and in Nahuatl, for instance, you pronounce the name of the Mexica tribe as meh-shee-kah). After the sound change, you'd think the spelling of places like Mexico or Oaxaca should also get a J. They did for a long time, actually, and Méjico is considered a valid spelling. Nonetheless, people get used to writing names a particular way, and don't tend to like to change them (sort of like how we write New York not New Yoick and New Orleans not Nawlins), and so the spelling with X was always used in Mexico itself, and today is the preferred spelling.

As a result, the letters X, J (and a G followed by E or I) will all sound like English's H.

Additionally, although not common for Mexican Spanish, you will notice that the S will often be pronounced like English's H. This generally happens when the S is at the end of a syllable, though not always.

^{1. In some imported words like hámster you will hear it aspirated, and in certain words in an extremely limited regions in Spain you can hear the h pronounced today, such that albahaca has an audible H.}